Understanding Limits: Solving Problems and Examples

[hamsterman]

I've been trying to learn some maths by myself. A book I found starts with a section on limits. I feel that I have a decent understanding of what is written, but then, there are some problems given that I just can't figure out. I feel like I'm missing something basic. I'm not sure what I'm looking for. Maybe a resource with some examples of how to solve different kinds of equations would be enough. I'd also appreciate it if you could show how to solve a couple of problems I'm having a hard time with:

$lim\sum\limits_{k=1}^{n-1} \frac{k^{2}}{n^{3}}, n\geq 2$

and

$lim\sum\limits_{k=2}^n \frac{k-1}{k!}, n\geq 2$

 

[mathman]

For the first one, take 1/n³ outside the summation, the sum over k² is readily available [ (n-1)n(2n-1)/6 ], so the limit will be 1/3.

For the second split it into two sums (k and -1 numerators). Compare them with each other. The final answer will be 1 (unless I made a mistake).

 

[hamsterman]

Thanks a lot, I see now.